{"paper":{"title":"Zero correlations and averaged fields of orthonormal Gaussian functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Iterated orthonormal Gaussian entire functions produce zero processes with index-dependent short-range correlations and averaged fields converging almost surely to 1.","cross_cats":["math-ph","math.CA","math.MP"],"primary_cat":"math.PR","authors_text":"Lu\\'is Daniel Abreu, Tomoyuki Shirai","submitted_at":"2026-05-17T07:16:21Z","abstract_excerpt":"We consider the family of point processes $\\{\\mathcal{Z}_{f_{n}}\\}_{n=0}^{\\infty}$ of zeros of Gaussian random functions $\\{f_{n}(z,\\overline{z})\\}_{n=0}^{\\infty} $, arising from the Gaussian Entire Function \\[ f_{0}(z):=\\sum_{k=0}^{\\infty} \\zeta_{k} \\frac{z^{k}}{\\sqrt{k!}}, \\quad \\zeta_{k} \\sim N_{\\mathbb{C}}(0,1)\\text{ i.i.d.} \\] by iteration of the Landau raising operator, and orthonormal at each point in expectation in the sense that \\[ \\mathbb{E}\\left[ e^{-\\left\\vert z\\right\\vert^{2}}f_{n}(z,\\overline{z})\\overline{f_{n^{\\prime }}(z,\\overline{z})}\\right] ={\\delta }_{nn'}. \\] We first show "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The normalized pair correlations g_{n,n+k}(z,w) exhibit repulsion for k=1, attraction for k=2, and no short-range second-order correlation for k >= 3 as w -> z; the averaged fields converge almost surely to 1 in C(K) and their scaled fluctuations converge to the Gaussian process G(z) = 1/sqrt(pi) int 1_{B(z,1)}(u) dW_R(u).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The sequence f_n is constructed by iterated application of the Landau raising operator to the base Gaussian entire function f_0 while preserving the pointwise orthonormality condition E[e^{-|z|^2} f_n(z, bar z) bar f_{n'}(z, bar z)] = delta_{n n'}.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Proves interlacing-like zero pair correlations (repulsion at k=1, attraction at k=2, none for k>=3) and a functional CLT for averaged fields of iterated Gaussian entire functions, confirming signal-processing conjectures.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Iterated orthonormal Gaussian entire functions produce zero processes with index-dependent short-range correlations and averaged fields converging almost surely to 1.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7bc59ce478aeee45e5f9ef7990572656f9725590a333aa8efb12e0985f09f65e"},"source":{"id":"2605.17296","kind":"arxiv","version":1},"verdict":{"id":"ee44e26e-86e0-466e-bdee-e2ca807451d3","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T23:12:13.062058Z","strongest_claim":"The normalized pair correlations g_{n,n+k}(z,w) exhibit repulsion for k=1, attraction for k=2, and no short-range second-order correlation for k >= 3 as w -> z; the averaged fields converge almost surely to 1 in C(K) and their scaled fluctuations converge to the Gaussian process G(z) = 1/sqrt(pi) int 1_{B(z,1)}(u) dW_R(u).","one_line_summary":"Proves interlacing-like zero pair correlations (repulsion at k=1, attraction at k=2, none for k>=3) and a functional CLT for averaged fields of iterated Gaussian entire functions, confirming signal-processing conjectures.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The sequence f_n is constructed by iterated application of the Landau raising operator to the base Gaussian entire function f_0 while preserving the pointwise orthonormality condition E[e^{-|z|^2} f_n(z, bar z) bar f_{n'}(z, bar z)] = delta_{n n'}.","pith_extraction_headline":"Iterated orthonormal Gaussian entire functions produce zero processes with index-dependent short-range correlations and averaged fields converging almost surely to 1."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17296/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T23:31:20.178012Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T23:21:25.512420Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T22:01:57.808085Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.763324Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"e9e8087c95cc2250b63743e09dd94ca3f76ea71d54112952d41731b6dd598044"},"references":{"count":72,"sample":[{"doi":"","year":2010,"title":"L. D. Abreu,Sampling and interpolation in Bargmann-Fock spaces of polyanalytic functions. Appl. Comp. Harm. Anal., 29, 287–302, (2010)","work_id":"a9610efc-d776-4032-848f-300110fbd7ba","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"L. D. Abreu, K. Gr¨ ochenig, J. L. Romero,On accumulated spectrograms. Trans. Amer. Math. Soc.368, 3629–3649, (2016)","work_id":"cd98b291-0055-455b-aedb-80f2e550ea4b","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2019,"title":"L. D. Abreu, K. Gr¨ ochenig, J. L. Romero,Harmonic analysis in phase space and finite Weyl-Heisenberg ensembles. J. Stat. Phys., vol. 174, 5, 1104–1136, (2019)","work_id":"021c7556-07ee-4502-98e4-8e1dc6f213eb","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"L. D. Abreu,Local maxima of white noise spectrograms and Gaussian Entire Functions. J. Fourier Anal. Appl., vol. 28, Article number: 88 (2022)","work_id":"377c97f7-1a44-4608-a050-60cb2706eacc","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"L. D. Abreu, D. Alpay, T. Georgiou, P. Jorgensen,Analytic continuation of time in Brownian motion. Stochastic distributions approach. J. Math. Anal. Appl., 130438, (2026)","work_id":"b6c3f6ea-9c2d-49b2-b855-cc000d210a97","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":72,"snapshot_sha256":"2b6adf7e604ba1da1dbcde14634a93e178644312da658f82b83997de0b5be057","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"2ddf4b981e6aabc53d30c13c679875e1b6c344237749720a75be186b3ee70e63"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}